EXAMPLE 2 Multiply rational expressions involving polynomials

Find the product 3x2 + 3x4x2 – 24x + 36

x2 – 4x + 3x2 – x

Multiply numerators and denominators.

3x2 + 3x4x2 – 24x + 36

x2 – 4x + 3x2 – x

=(3x2 + 3x) (x2 – 4x + 3)(4x2 – 24x + 36)(x2 – x)

Factor and divide out common factors.

= 3(x + 1)4(x – 3)

Simplify.

= 3x(x + 1)(x – 3)(x – 1)4x(x – 3)(x – 3)(x – 1)

EXAMPLE 2 Multiply rational expressions involving polynomials

CHECK

Check your simplification using a graphing calculator.

The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3).

Graph y1 = 3x2 + 3x4x2 – 24x + 36

x2 – 4x + 3x2 – x

and y2 =3(x + 1)4(x – 3)

.

EXAMPLE 3 Multiply a rational expression by a polynomial

Find the product 5x

x2 + 5x + 6(x + 3).

5x

x2 + 5x + 6(x + 3).

5x

x2 + 5x + 6

(x + 3)=

1

Rewrite polynomial as a fraction.

= 5x(x + 3)x2 + 5x + 6

Multiply numerators and denominators.

=5x(x + 3)

(x + 3)(x + 2)

Factor and divide out common factor.

Simplify.=5x

x + 2

GUIDED PRACTICE for Examples 2 and 3

3. Find the product x2 + x – 2 x2 + x

2x2 + 2x5x2 –15x +10

2(x + 1)5(x – 2)

ANSWER

Find the product2w2

w2 – 7w + 12(w – 4).4.

2w2

w – 3ANSWER